Measuring the proper motions and geometric distances of galaxies within the Local Group is very important for our understanding of the history , present state and future of the Local Group .
Currently , proper motion measurements using optical methods are limited only to the closest companions of the Milky Way .
However , Very Long Baseline Interferometry ( VLBI ) provides the best angular resolution in astronomy and phase-referencing techniques yield astrometric accuracies of \approx 10 micro-arcseconds .
This makes a measurement of proper motions and angular rotation rates of galaxies out to a distance of \sim 1 Mpc feasible .

This article presents results of VLBI observations of regions of H _ { 2 } O maser activity in the Local Group galaxies M33 and IC 10 .
Two masing regions in M33 are on opposite sides of the galaxy .
This allows a comparison of the angular rotation rate ( as measured by the VLBI observations ) with the known inclination and rotation speed of the HI gas disk .
This gives a geometric distance of 730 \pm 100 \pm 135 kpc .
The first error indicates the statistical error from the proper motion measurements while the second error is the systematic error from the rotation model .
This distance is consistent , within the errors , with the most recent Cepheid distance to M33 .
Since all position measurements were made relative to an extragalactic background source , the proper motion of M33 has also been measured .
This provides a three dimensional velocity vector of M33 , showing that this galaxy is moving with a velocity of 190 \pm 59 km s ^ { -1 } relative to the Milky Way .
For IC 10 , we obtain a motion of 215 \pm 42 km s ^ { -1 } relative to the Milky Way .
These measurements promise a new handle on dynamical models for the Local Group and the mass and dark matter halo of Andromeda and the Milky Way .